Louis Nirenberg facts for kids
Quick facts for kids
Louis Nirenberg
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Nirenberg in 1975
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| Born | February 28, 1925 Hamilton, Ontario, Canada
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| Died | January 26, 2020 (aged 94) Manhattan, New York, U.S.
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| Citizenship | Canadian and American |
| Alma mater | McGill University (BS, 1945) New York University (PhD, 1950) |
| Known for | Partial differential equations Gagliardo–Nirenberg interpolation inequality Gagliardo–Nirenberg–Sobolev inequality Bounded mean oscillation (John–Nirenberg space) Nirenberg's conjecture |
| Awards | Bôcher Memorial Prize (1959) Crafoord Prize (1982) Steele Prize (1994, 2014) National Medal of Science (1995) Chern Medal (2010) Abel Prize in Mathematics (2015) |
| Scientific career | |
| Fields | Mathematics |
| Institutions | New York University |
| Thesis | The determination of a closed convex surface having given line elements (1949) |
| Doctoral advisor | James Stoker |
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Courant Institute of Mathematical Sciences
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Louis Nirenberg (February 28, 1925 – January 26, 2020) was a brilliant Canadian-American mathematician. Many people consider him one of the most important mathematicians of the 20th century.
Most of his work focused on partial differential equations. These are special math equations that describe how things change. His ideas are now basic to this field. For example, his "strong maximum principle" helps understand how solutions to these equations behave. He also helped create the Newlander-Nirenberg theorem in complex geometry. Nirenberg is seen as a key person in an area called geometric analysis. His work often connected to complex analysis and differential geometry.
Biography: Louis Nirenberg's Life Story
Louis Nirenberg was born in Hamilton, Ontario, Canada. His parents were immigrants from Ukraine. He went to Baron Byng High School and then McGill University. In 1945, he earned his Bachelor of Science degree in both math and physics.
During a summer job, he met Sara Paul, whose father was the famous mathematician Richard Courant. She helped Nirenberg get into graduate school at the Courant Institute of Mathematical Sciences at New York University. In 1949, he earned his PhD in mathematics. His advisor was James Stoker. For his PhD, Nirenberg solved a tough math problem called the "Weyl problem." This problem had been unsolved since 1916.
After getting his doctorate, he became a professor at the Courant Institute. He stayed there for his entire career. He guided 45 students through their PhD studies. He also wrote over 150 math papers with many co-authors. He kept doing math research until he was 87 years old. Louis Nirenberg passed away on January 26, 2020, at the age of 94.
Nirenberg received many important awards for his work. These include:
- Bôcher Memorial Prize (1959)
- Elected to the American Academy of Arts and Sciences (1965)
- Elected to the United States National Academy of Sciences (1969)
- Crafoord Prize (1982)
- Jeffery–Williams Prize (1987)
- Elected to the American Philosophical Society (1987)
- Steele Prize for Lifetime Achievement (1994)
- National Medal of Science (1995)
- Chern Medal (2010)
- Steele Prize for Seminal Contribution to Research (2014) with Luis Caffarelli and Robert Kohn
- Abel Prize (2015)
Louis Nirenberg's Math Discoveries
Nirenberg is well-known for his work with Shmuel Agmon and Avron Douglis. They made a math theory called "Schauder theory" much broader. This theory helps understand certain types of math equations. With Basilis Gidas and Wei-Ming Ni, he used the "maximum principle" in new ways. This helped prove that many solutions to differential equations have a special kind of symmetry.
Nirenberg and Fritz John started studying the BMO function space in 1961. This space was first used to study elastic materials. But it has also been used in areas like games of chance. In 1982, Nirenberg, Luis Caffarelli, and Robert Kohn made a big step forward. Their work helped understand the Navier–Stokes existence and smoothness equations. These equations are used to describe how fluids move.
Other important achievements by Nirenberg include:
- Solving the Minkowski problem in two dimensions.
- Developing the Gagliardo–Nirenberg interpolation inequality.
- Creating the Newlander-Nirenberg theorem in complex geometry.
- Helping to develop "pseudo-differential operators" with Joseph Kohn.
The Navier–Stokes equations were created in the early 1800s. They help us understand how fluids like water or air move. In the 1930s, Jean Leray found a way to show that solutions to these equations exist.
A big step happened in the 1970s with Vladimir Scheffer's work. He showed that if a smooth solution to the Navier-Stokes equations gets to a "singular time" (a point where it might break down), it can still be understood. Soon after, Luis Caffarelli, Robert Kohn, and Nirenberg made Scheffer's ideas even better. Their work helped explain how solutions to these equations stay "smooth" (well-behaved) in most places.
In 2014, the American Mathematical Society gave the Steele Prize to Caffarelli, Kohn, and Nirenberg for their paper. They called it a "landmark" work that inspired many mathematicians. Understanding the Navier-Stokes equations fully is still a big challenge in math today.
Nonlinear Elliptic Equations
In the 1930s, Charles Morrey studied a type of math problem called "quasilinear elliptic partial differential equations." Nirenberg, in his PhD work, made Morrey's results much broader. He applied them to "fully nonlinear elliptic equations." Morrey and Nirenberg's early work mostly focused on two-dimensional problems. Understanding these equations in higher dimensions was a big open question.
The Monge-Ampère equation is a key example of a fully nonlinear elliptic equation. Nirenberg worked with Eugenio Calabi on this equation. Later, with Luis Caffarelli and Joel Spruck, Nirenberg found ways to solve these problems. They showed how to find solutions and understand their behavior near boundaries. Their methods helped solve many complex math problems.
Linear Elliptic Systems
One of Nirenberg's most famous works from the 1950s is about "elliptic regularity." With Avron Douglis, Nirenberg expanded the "Schauder estimates." These estimates help us understand how smooth the solutions to elliptic equations are. They applied these ideas to more general systems of equations.
Working with Shmuel Agmon and Douglis, Nirenberg also proved how solutions behave near the edges of a problem's area. These discoveries are now standard tools for mathematicians. The Douglis-Nirenberg and Agmon-Douglis-Nirenberg estimates are among the most used tools in the study of partial differential equations.
With Yanyan Li, Nirenberg studied equations where the properties of materials change suddenly. They found that the "gradient" (how fast something changes) of the solution is smooth. This was true even when the material properties were not smooth at the edges.
Maximum Principle and Its Uses
The "maximum principle" is a basic idea in math. It was known for some types of equations in the 1800s. Nirenberg made this principle apply to a new type of equation called "parabolic partial differential equations." This was one of his first important works. His discovery is now a core part of understanding these equations.
In the 1950s, A.D. Alexandrov introduced a clever "moving plane" method. This method helps prove that certain shapes, like a perfect sphere, are unique. Nirenberg saw that this method could be used to show that solutions to certain equations often have the same symmetry as the problem itself.
Nirenberg, Basilis Gidas, and Wei-Ming Ni developed this technique further. They showed that solutions to many nonlinear equations have rotational symmetry. This means that instead of studying complex partial differential equations, mathematicians could sometimes study simpler "ordinary differential equations."
Later, with Henri Berestycki, Nirenberg improved these methods. They also worked with Srinivasa Varadhan to study how the maximum principle applies in more general situations. Their work helped solve a famous math problem about "translational symmetry."
Functional Inequalities
Nirenberg and Emilio Gagliardo independently discovered important math rules called "Gagliardo–Nirenberg–Sobolev inequality" and "Gagliardo–Nirenberg interpolation inequalities." These rules are used all the time in the study of partial differential equations. They help mathematicians understand how different math spaces relate to each other.
After Fritz John introduced the BMO function space, he and Nirenberg studied it more deeply. They proved the "John–Nirenberg inequality." This inequality helps understand how much a BMO function can differ from its average value. Their work is now a standard part of many math textbooks. It is used in areas like probability, complex analysis, and Fourier analysis.
Nirenberg and Haïm Brézis also applied the BMO theory to maps between complex shapes. They showed that maps that are "close" in BMO behave similarly. This helped define "topological degree" for certain types of maps.
Calculus of Variations
In math, "calculus of variations" is about finding functions that make certain quantities as small or as large as possible. Nirenberg worked with Haïm Brezis and Guido Stampacchia on this topic. Their work has uses in an area called "variational inequalities."
Nirenberg, Brezis, and Jean-Michel Coron found a new way to study "wave equations." They used a special math tool called the "mountain pass theorem." This helped them find solutions that repeat over time. Their work made the mountain pass theorem even more useful for other mathematicians.
A key idea from Brezis and Nirenberg was about "local minimizers." They showed that sometimes, if a function is a minimizer in a simpler group of functions, it's also a minimizer in a larger, more complex group.
In one of his most cited papers, Nirenberg and Brézis studied a type of equation called "Yamabe-type equations." With Berestycki and Italo Capuzzo-Dolcetta, Nirenberg also studied "superlinear equations." This work helped understand when solutions to these equations exist or do not exist.
Geometric Problems
Nirenberg's PhD thesis helped solve the "Weyl problem" and the "Minkowski problem" in differential geometry. The Weyl problem asks if you can smoothly embed a curved surface into 3D space. The Minkowski problem asks if you can find a closed surface based on how its curvature is given. Nirenberg's work showed that certain equations from surface theory could be solved using his new math tools.
In one of his few papers not focused on pure analysis, Nirenberg and Philip Hartman described cylinders in Euclidean space. They showed that cylinders are the only complete "hypersurfaces" that are "intrinsically flat." This means they are flat when you measure distances on their surface.
Nirenberg and his student August Newlander proved the Newlander-Nirenberg theorem. This theorem gives a clear math condition for when a special structure on a manifold (a type of geometric space) can be described using complex coordinates. This theorem is now a basic result in complex geometry.
Nirenberg and Charles Loewner studied how to give a "complete Riemannian metric" to bounded open sets in Euclidean space. This means finding a natural way to measure distances within these spaces. They showed that solutions to certain equations could define metrics with constant curvature.
Pseudo-differential Operators
Joseph Kohn and Nirenberg introduced the idea of "pseudo-differential operators." These are like generalized versions of differential operators. Nirenberg and François Trèves studied a famous example by Hans Lewy. This example showed a linear partial differential equation that couldn't be solved. Nirenberg and Trèves found the conditions under which such equations *can* be solved. Their work opened new doors for understanding how to solve linear partial differential equations.
See also
In Spanish: Louis Nirenberg para niños
